Lectures on String Theory

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Exercise 10.
• Show that the constraints
have the following Poisson brackets
C 1 = P µ P µ + T 2 X ′ µX ′µ , C 2 = P µ X ′µ
{C 1 (σ), C 1 (σ ′ )} = 4T 2 ∂ σ C 2 (σ)δ(σ − σ ′ ) + 8T 2 C 2 (σ)∂ σ δ(σ − σ ′ ) ,
{C 1 (σ), C 2 (σ ′ )} = ∂ σ C 1 (σ)δ(σ − σ ′ ) + 2C 1 (σ)∂ σ δ(σ − σ ′ ) ,
{C 2 (σ), C 1 (σ ′ )} = ∂ σ C 1 (σ)δ(σ − σ ′ ) + 2C 1 (σ)∂ σ δ(σ − σ ′ ) ,
{C 2 (σ), C 2 (σ ′ )} = ∂ σ C 2 (σ)δ(σ − σ ′ ) + 2C 2 (σ)∂ σ δ(σ − σ ′ ) .
• Define the linear combinations
T ++ = 1
8T 2 (C 1 + 2T C 2 ) = 1
8T 2 (P µ + T X ′ µ) 2 ,
T −− = 1
8T 2 (C 1 − 2T C 2 ) = 1
8T 2 (P µ − T X ′ µ) 2 .
and show that their Poisson algebra is
{T ++ (σ), T ++ (σ ′ )} = 1 (
)
∂ σ T ++ (σ)δ(σ − σ ′ ) + 2T ++ (σ)∂ σ δ(σ − σ ′ ) ,
2T
{T −− (σ), T −− (σ ′ )} = − 1 (
)
∂ σ T −− (σ)δ(σ − σ ′ ) + 2T −− (σ)∂ σ δ(σ − σ ′ ) ,
2T
{T ++ (σ), T −− (σ ′ )} = 0 .
Exercise 11. For the closed string case define
L m = 2T
¯L m = 2T
∫ 2π
0
∫ 2π
0
dσ e imσ− T −− (σ, τ)
dσ e imσ+ T ++ (σ, τ) .
Show that for any integer m the generators L m and ¯L m are time-independent.
Exercise 12. Compute the Poisson brackets of the constraints L m , ¯L m . What
kind of constraints they are, i.e. the first or the second class?
Exercise 13. It is known in curved space-time that we can transform the metric
locally in the neighborhood of a point x µ = 0 to the following form g µν (x) = η µν −